Prove that x\u00b2+x+1 is always positive.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To prove that ( x^2 + x + 1 ) is always positive, we can use algebraic techniques, such as analyzing its discriminant and considering its behavior for all values of ( x ).<\/p>\n\n\n\n We are looking at the quadratic function ( f(x) = x^2 + x + 1 ), which is a parabola opening upwards because the coefficient of ( x^2 ) is positive.<\/p>\n\n\n\n The discriminant of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( \\Delta = b^2 – 4ac ). For our equation, ( a = 1 ), ( b = 1 ), and ( c = 1 ), so the discriminant is:<\/p>\n\n\n\n [ Since the discriminant is negative (( \\Delta = -3 )), the quadratic equation has no real roots. This means that the function ( f(x) = x^2 + x + 1 ) does not cross the x-axis and does not take negative values for any real value of ( x ).<\/p>\n\n\n\n Since the quadratic opens upwards (because the leading coefficient is positive), the vertex of the parabola gives the minimum value of the function. The x-coordinate of the vertex is given by:<\/p>\n\n\n\n [ Substitute ( x = -\\frac{1}{2} ) into the function to find the minimum value:<\/p>\n\n\n\n [ Thus, the minimum value of ( f(x) = x^2 + x + 1 ) is ( \\frac{3}{4} ), which is positive.<\/p>\n\n\n\n Since the quadratic function ( f(x) = x^2 + x + 1 ) has no real roots, opens upwards, and has a minimum value of ( \\frac{3}{4} ), it is always positive for all real values of ( x ). Therefore, we have proven that ( x^2 + x + 1 ) is always positive.<\/p>\n","protected":false},"excerpt":{"rendered":" Prove that x\u00b2+x+1 is always positive. The Correct Answer and Explanation is : To prove that ( x^2 + x + 1 ) is always positive, we can use algebraic techniques, such as analyzing its discriminant and considering its behavior for all values of ( x ). Step 1: Consider the function We are looking […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-163591","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163591","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=163591"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163591\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=163591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=163591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=163591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Consider the function<\/h3>\n\n\n\n
Step 2: Calculate the discriminant<\/h3>\n\n\n\n
\\Delta = 1^2 – 4(1)(1) = 1 – 4 = -3
]<\/p>\n\n\n\nStep 3: Analyze the vertex<\/h3>\n\n\n\n
x = \\frac{-b}{2a} = \\frac{-1}{2(1)} = -\\frac{1}{2}
]<\/p>\n\n\n\n
f\\left(-\\frac{1}{2}\\right) = \\left(-\\frac{1}{2}\\right)^2 + \\left(-\\frac{1}{2}\\right) + 1 = \\frac{1}{4} – \\frac{1}{2} + 1 = \\frac{3}{4}
]<\/p>\n\n\n\nStep 4: Conclusion<\/h3>\n\n\n\n